UPDATE 1-Sterling slips after UK service sector growth slows

* Graphic: World FX rates in 2018 http://tmsnrt.rs/2egbfVh

Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv (Updates after PMI data published, adds details)

LONDON, April 5 (Reuters) - Sterling slipped further against the dollar and the euro on Thursday after a major survey showed British services growing at its slowest rate since just after the vote to exit the European Union in June 2016.

Heavy snow - brought by the "Beast from the East" cold weather bout - and weak consumer demand weighed on British services in March, with the IHS Markit/CIPS services Purchasing Managers' Index (PMI) tumbling to a worse-than-expected 51.7 from February's reading of 54.5.

Britain's construction PMI showed a similar drop on Wednesday, although manufacturing held up better.

Traders this week have largely shrugged off the data as they focus on any signs economic weakness could steer the Bank of England from its path of an expected interest rate rise in May.

Jordan Rochester, an FX strategist at Nomura, said the pound's reaction to the data was only marginal as the survey showed input prices rising, which pointed to inflation remaining above target and the BoE needing to act.

"It won't have much of an impact on the May BoE hike. As a policymaker you would look through the weather-related weakness," he said.

The pound, which was already down 0.2 percent, fell as much as 0.3 percent to $1.4030 after the data.

Against the euro, sterling fell 0.2 percent to 87.380 pence per euro before recovering slightly.

"Fingers will point to the Beast from the East, but my sense is longer term underlying issues remain in play, especially given this is the weakest number since the Brexit vote," said Neil Jones, head of FX hedge fund sales at Mizuho bank.

Sterling's fall on Thursday came as the dollar rebounded on an ease in global trade war fears and investors paused ahead of a survey of Britain's crucial services sector.

The predicted rate rise in May is largely baked into sterling's price, and combined with Britain agreeing a Brexit transition deal last month and dollar weakness caused by a growing trade dispute with China have kept the pound above $1.40. (Reporting by Tommy Wilkes; Editing by Angus MacSwan)